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PROPERTIES OF THE INVERSE OF A NONCENTRAL WISHART MATRIX

Published online by Cambridge University Press:  27 May 2021

Grant Hillier  and
Raymond Kan
Grant Hillier*
Affiliation:
University of Southampton
Raymond Kan
Affiliation:
University of Toronto
*
Address correspondence to Grant Hillier, Economics Division, School of Social Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK; e-mail: ghh@soton.ac.uk.
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Abstract

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The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 6: YALE 2018 CONFERENCE IN HONOR OF PETER C. B. PHILLIPS: PART II , December 2022 , pp. 1092 - 1116
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Thanks to the Editor, Co-Editor, and two referees for helpful comments on the first version of the paper.

References

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